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Theorem ipasslem1 25408
Description: Lemma for ipassi 25418. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  U )
ip1i.9  |-  U  e.  CPreHil
OLD
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )

Proof of Theorem ipasslem1
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0cn 10794 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  CC )
2 ax-1cn 9539 . . . . . . . . . . . 12  |-  1  e.  CC
3 ip1i.9 . . . . . . . . . . . . . 14  |-  U  e.  CPreHil
OLD
43phnvi 25393 . . . . . . . . . . . . 13  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . . . . . . . . . 14  |-  X  =  ( BaseSet `  U )
6 ip1i.2 . . . . . . . . . . . . . 14  |-  G  =  ( +v `  U
)
7 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .sOLD `  U )
85, 6, 7nvdir 25188 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
k  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
94, 8mpan 670 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  X )  ->  (
( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
102, 9mp3an2 1307 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
111, 10sylan 471 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
125, 7nvsid 25184 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
134, 12mpan 670 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  (
1 S A )  =  A )
1413adantl 466 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1514oveq2d 6291 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k S A ) G ( 1 S A ) )  =  ( ( k S A ) G A ) )
1611, 15eqtrd 2501 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G A ) )
1716oveq1d 6290 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) G A ) P B ) )
18 ipasslem1.b . . . . . . . . . . . . 13  |-  B  e.  X
19 ip1i.7 . . . . . . . . . . . . . 14  |-  P  =  ( .iOLD `  U )
205, 19dipcl 25287 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
214, 18, 20mp3an13 1310 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
2221mulid2d 9603 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2322adantl 466 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1  x.  ( A P B ) )  =  ( A P B ) )
2423oveq2d 6291 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
255, 7nvscl 25183 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  k  e.  CC  /\  A  e.  X )  ->  (
k S A )  e.  X )
264, 25mp3an1 1306 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( k S A )  e.  X )
271, 26sylan 471 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( k S A )  e.  X )
285, 6, 7, 19, 3ipdiri 25407 . . . . . . . . . . 11  |-  ( ( ( k S A )  e.  X  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
2918, 28mp3an3 1308 . . . . . . . . . 10  |-  ( ( ( k S A )  e.  X  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3027, 29sylancom 667 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3124, 30eqtr4d 2504 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) G A ) P B ) )
3217, 31eqtr4d 2504 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) ) )
33 oveq1 6282 . . . . . . 7  |-  ( ( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  ->  (
( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3432, 33sylan9eq 2521 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
35 adddir 9576 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
362, 35mp3an2 1307 . . . . . . . 8  |-  ( ( k  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
371, 21, 36syl2an 477 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3837adantr 465 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3934, 38eqtr4d 2504 . . . . 5  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
4039exp31 604 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  X  ->  (
( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  -> 
( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
4140a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  X  -> 
( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( A  e.  X  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
42 eqid 2460 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
435, 42, 19dip0l 25293 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
444, 18, 43mp2an 672 . . . 4  |-  ( (
0vec `  U ) P B )  =  0
455, 7, 42nv0 25194 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  ( 0vec `  U
) )
464, 45mpan 670 . . . . 5  |-  ( A  e.  X  ->  (
0 S A )  =  ( 0vec `  U
) )
4746oveq1d 6290 . . . 4  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( ( 0vec `  U ) P B ) )
4821mul02d 9766 . . . 4  |-  ( A  e.  X  ->  (
0  x.  ( A P B ) )  =  0 )
4944, 47, 483eqtr4a 2527 . . 3  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) )
50 oveq1 6282 . . . . . 6  |-  ( j  =  0  ->  (
j S A )  =  ( 0 S A ) )
5150oveq1d 6290 . . . . 5  |-  ( j  =  0  ->  (
( j S A ) P B )  =  ( ( 0 S A ) P B ) )
52 oveq1 6282 . . . . 5  |-  ( j  =  0  ->  (
j  x.  ( A P B ) )  =  ( 0  x.  ( A P B ) ) )
5351, 52eqeq12d 2482 . . . 4  |-  ( j  =  0  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) )
5453imbi2d 316 . . 3  |-  ( j  =  0  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) ) )
55 oveq1 6282 . . . . . 6  |-  ( j  =  k  ->  (
j S A )  =  ( k S A ) )
5655oveq1d 6290 . . . . 5  |-  ( j  =  k  ->  (
( j S A ) P B )  =  ( ( k S A ) P B ) )
57 oveq1 6282 . . . . 5  |-  ( j  =  k  ->  (
j  x.  ( A P B ) )  =  ( k  x.  ( A P B ) ) )
5856, 57eqeq12d 2482 . . . 4  |-  ( j  =  k  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
k S A ) P B )  =  ( k  x.  ( A P B ) ) ) )
5958imbi2d 316 . . 3  |-  ( j  =  k  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) ) ) )
60 oveq1 6282 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j S A )  =  ( ( k  +  1 ) S A ) )
6160oveq1d 6290 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j S A ) P B )  =  ( ( ( k  +  1 ) S A ) P B ) )
62 oveq1 6282 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  ( A P B ) )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
6361, 62eqeq12d 2482 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) )
6463imbi2d 316 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
65 oveq1 6282 . . . . . 6  |-  ( j  =  N  ->  (
j S A )  =  ( N S A ) )
6665oveq1d 6290 . . . . 5  |-  ( j  =  N  ->  (
( j S A ) P B )  =  ( ( N S A ) P B ) )
67 oveq1 6282 . . . . 5  |-  ( j  =  N  ->  (
j  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
6866, 67eqeq12d 2482 . . . 4  |-  ( j  =  N  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
6968imbi2d 316 . . 3  |-  ( j  =  N  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) ) )
7041, 49, 54, 59, 64, 69nn0indALT 10947 . 2  |-  ( N  e.  NN0  ->  ( A  e.  X  ->  (
( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
7170imp 429 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NN0cn0 10784   NrmCVeccnv 25139   +vcpv 25140   BaseSetcba 25141   .sOLDcns 25142   0veccn0v 25143   .iOLDcdip 25272   CPreHil OLDccphlo 25389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-grpo 24855  df-gid 24856  df-ginv 24857  df-ablo 24946  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-nmcv 25155  df-dip 25273  df-ph 25390
This theorem is referenced by:  ipasslem2  25409  ipasslem3  25410  ipasslem4  25411
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